3.32.32 \(\int \frac {\sqrt {-b+a^2 x^2}}{\sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx\)

Optimal. Leaf size=803 \[ -\frac {49725 \tan ^{-1}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {a^2 x^2-b}}}}{\sqrt [4]{c}}\right ) b^2}{524288 a c^{29/4}}+\frac {49725 \tanh ^{-1}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {a^2 x^2-b}}}}{\sqrt [4]{c}}\right ) b^2}{524288 a c^{29/4}}+\frac {3 \tan ^{-1}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {a^2 x^2-b}}}}{\sqrt [4]{c}}\right ) b}{4 a c^{5/4}}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {a^2 x^2-b}}}}{\sqrt [4]{c}}\right ) b}{4 a c^{5/4}}+\frac {\left (a x+\sqrt {a^2 x^2-b}\right )^{2/3} \left (c+\sqrt [3]{a x+\sqrt {a^2 x^2-b}}\right )^{3/4} \left (-805306368 a^2 x^2 c^{10}[phone] b c^{10}[phone] a^3 x^3 c^7-908328960 a b x c^7-224716800 b^2 c^4+290990700 a b^2 x c\right )+\sqrt {a^2 x^2-b} \left (\left (a x+\sqrt {a^2 x^2-b}\right )^{2/3} \left (c+\sqrt [3]{a x+\sqrt {a^2 x^2-b}}\right )^{3/4} \left (-805306368 a x c^{10}[phone] a^2 x^2 c^7-302776320 b c^7+290990700 b^2 c\right )+\left (1409286144 a^2 x^2 c^9-352321536 b c^9+5752750080 a b x c^6+238761600 b^2 c^3-727476750 a b^2 x\right ) \left (c+\sqrt [3]{a x+\sqrt {a^2 x^2-b}}\right )^{3/4}+\left (1073741824 a x c^{11}-1291845632 a^2 x^2 c^8+322961408 b c^8-258658400 b^2 c^2\right ) \sqrt [3]{a x+\sqrt {a^2 x^2-b}} \left (c+\sqrt [3]{a x+\sqrt {a^2 x^2-b}}\right )^{3/4}\right )+\left (1409286144 a^3 x^3 c^9-1056964608 a b x c^9-3081830400 b^2 c^6+5752750080 a^2 b x^2 c^6+238761600 a b^2 x c^3+363738375 b^3-727476750 a^2 b^2 x^2\right ) \left (c+\sqrt [3]{a x+\sqrt {a^2 x^2-b}}\right )^{3/4}+\left (1073741824 a^2 x^2 c^{11}-536870912 b c^{11}-1291845632 a^3 x^3 c^8+968884224 a b x c^8+214016000 b^2 c^5-258658400 a b^2 x c^2\right ) \sqrt [3]{a x+\sqrt {a^2 x^2-b}} \left (c+\sqrt [3]{a x+\sqrt {a^2 x^2-b}}\right )^{3/4}}{1917583360 a c^7 \left (a x+\sqrt {a^2 x^2-b}\right )^{7/3}} \]

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Rubi [F]  time = 1.13, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\sqrt {-b+a^2 x^2}}{\sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[Sqrt[-b + a^2*x^2]/((a*x + Sqrt[-b + a^2*x^2])^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)),x]

[Out]

Defer[Int][Sqrt[-b + a^2*x^2]/((a*x + Sqrt[-b + a^2*x^2])^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)),
x]

Rubi steps

\begin {align*} \int \frac {\sqrt {-b+a^2 x^2}}{\sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx &=\int \frac {\sqrt {-b+a^2 x^2}}{\sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx\\ \end {align*}

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Mathematica [F]  time = 180.00, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[Sqrt[-b + a^2*x^2]/((a*x + Sqrt[-b + a^2*x^2])^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)),x
]

[Out]

$Aborted

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IntegrateAlgebraic [A]  time = 2.45, size = 803, normalized size = 1.00 \begin {gather*} \frac {\left (363738375 b^3-3081830400 b^2 c^6+238761600 a b^2 c^3 x-1056964608 a b c^9 x-727476750 a^2 b^2 x^2+5752750080 a^2 b c^6 x^2+1409286144 a^3 c^9 x^3\right ) \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\left (214016000 b^2 c^5-536870912 b c^{11}-258658400 a b^2 c^2 x+968884224 a b c^8 x+1073741824 a^2 c^{11} x^2-1291845632 a^3 c^8 x^3\right ) \sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\left (-224716800 b^2 c^4+402653184 b c^{10}[phone] a b^2 c x-908328960 a b c^7 x-805306368 a^2 c^{10} x^2+1211105280 a^3 c^7 x^3\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\sqrt {-b+a^2 x^2} \left (\left (238761600 b^2 c^3-352321536 b c^9-727476750 a b^2 x+5752750080 a b c^6 x+1409286144 a^2 c^9 x^2\right ) \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\left (-258658400 b^2 c^2+322961408 b c^8+1073741824 a c^{11} x-1291845632 a^2 c^8 x^2\right ) \sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\left (290990700 b^2 c-302776320 b c^7-805306368 a c^{10} x+1211105280 a^2 c^7 x^2\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}\right )}{1917583360 a c^7 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/3}}-\frac {49725 b^2 \tan ^{-1}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{524288 a c^{29/4}}+\frac {3 b \tan ^{-1}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{4 a c^{5/4}}+\frac {49725 b^2 \tanh ^{-1}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{524288 a c^{29/4}}-\frac {3 b \tanh ^{-1}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{4 a c^{5/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[-b + a^2*x^2]/((a*x + Sqrt[-b + a^2*x^2])^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))
^(1/4)),x]

[Out]

((363738375*b^3 - 3081830400*b^2*c^6 + 238761600*a*b^2*c^3*x - 1056964608*a*b*c^9*x - 727476750*a^2*b^2*x^2 +
5752750080*a^2*b*c^6*x^2 + 1409286144*a^3*c^9*x^3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4) + (214016000*b
^2*c^5 - 536870912*b*c^11 - 258658400*a*b^2*c^2*x + 968884224*a*b*c^8*x + 1073741824*a^2*c^11*x^2 - 1291845632
*a^3*c^8*x^3)*(a*x + Sqrt[-b + a^2*x^2])^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4) + (-224716800*b^2*
c^4 + 402653184*b*c^10 + 290990700*a*b^2*c*x - 908328960*a*b*c^7*x - 805306368*a^2*c^10*x^2 + 1211105280*a^3*c
^7*x^3)*(a*x + Sqrt[-b + a^2*x^2])^(2/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4) + Sqrt[-b + a^2*x^2]*((2
38761600*b^2*c^3 - 352321536*b*c^9 - 727476750*a*b^2*x + 5752750080*a*b*c^6*x + 1409286144*a^2*c^9*x^2)*(c + (
a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4) + (-258658400*b^2*c^2 + 322961408*b*c^8 + 1073741824*a*c^11*x - 1291845
632*a^2*c^8*x^2)*(a*x + Sqrt[-b + a^2*x^2])^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4) + (290990700*b^
2*c - 302776320*b*c^7 - 805306368*a*c^10*x + 1211105280*a^2*c^7*x^2)*(a*x + Sqrt[-b + a^2*x^2])^(2/3)*(c + (a*
x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4)))/(1917583360*a*c^7*(a*x + Sqrt[-b + a^2*x^2])^(7/3)) - (49725*b^2*ArcTan
[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)/c^(1/4)])/(524288*a*c^(29/4)) + (3*b*ArcTan[(c + (a*x + Sqrt[-b
+ a^2*x^2])^(1/3))^(1/4)/c^(1/4)])/(4*a*c^(5/4)) + (49725*b^2*ArcTanh[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(
1/4)/c^(1/4)])/(524288*a*c^(29/4)) - (3*b*ArcTanh[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)/c^(1/4)])/(4*a*
c^(5/4))

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fricas [A]  time = 0.73, size = 1121, normalized size = 1.40

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*x^2-b)^(1/2)/(a*x+(a^2*x^2-b)^(1/2))^(1/3)/(c+(a*x+(a^2*x^2-b)^(1/2))^(1/3))^(1/4),x, algorithm
="fricas")

[Out]

-1/7670333440*(87780*a*b*c^7*((295147905179352825856*b^4*c^24 - 149294327647331942400*b^5*c^18 + 2831901719003
1360000*b^6*c^12 - 2387429351424000000*b^7*c^6 + 75476916312890625*b^8)/(a^4*c^29))^(1/4)*arctan((sqrt((507060
2400912917605986812821504*b^6*c^36 - 3847285528341595885584934502400*b^7*c^30 + 121629252686046508414402560000
0*b^8*c^24 - 205078619717531420590080000000*b^9*c^18 + 19450253230007648256000000000*b^10*c^12 - 9838499367900
90240000000000*b^11*c^6 + 20735820391713136962890625*b^12)*sqrt(c + (a*x + sqrt(a^2*x^2 - b))^(1/3)) + (295147
905179352825856*a^2*b^4*c^39 - 149294327647331942400*a^2*b^5*c^33 + 28319017190031360000*a^2*b^6*c^27 - 238742
9351424000000*a^2*b^7*c^21 + 75476916312890625*a^2*b^8*c^15)*sqrt((295147905179352825856*b^4*c^24 - 1492943276
47331942400*b^5*c^18 + 28319017190031360000*b^6*c^12 - 2387429351424000000*b^7*c^6 + 75476916312890625*b^8)/(a
^4*c^29)))*a*c^7*((295147905179352825856*b^4*c^24 - 149294327647331942400*b^5*c^18 + 28319017190031360000*b^6*
c^12 - 2387429351424000000*b^7*c^6 + 75476916312890625*b^8)/(a^4*c^29))^(1/4) - (2251799813685248*a*b^3*c^25 -
854268995174400*a*b^4*c^19 + 108028477440000*a*b^5*c^13 - 4553660109375*a*b^6*c^7)*(c + (a*x + sqrt(a^2*x^2 -
b))^(1/3))^(1/4)*((295147905179352825856*b^4*c^24 - 149294327647331942400*b^5*c^18 + 28319017190031360000*b^6
*c^12 - 2387429351424000000*b^7*c^6 + 75476916312890625*b^8)/(a^4*c^29))^(1/4))/(295147905179352825856*b^4*c^2
4 - 149294327647331942400*b^5*c^18 + 28319017190031360000*b^6*c^12 - 2387429351424000000*b^7*c^6 + 75476916312
890625*b^8)) + 21945*a*b*c^7*((295147905179352825856*b^4*c^24 - 149294327647331942400*b^5*c^18 + 2831901719003
1360000*b^6*c^12 - 2387429351424000000*b^7*c^6 + 75476916312890625*b^8)/(a^4*c^29))^(1/4)*log(27*a^3*c^22*((29
5147905179352825856*b^4*c^24 - 149294327647331942400*b^5*c^18 + 28319017190031360000*b^6*c^12 - 23874293514240
00000*b^7*c^6 + 75476916312890625*b^8)/(a^4*c^29))^(3/4) + 27*(2251799813685248*b^3*c^18 - 854268995174400*b^4
*c^12 + 108028477440000*b^5*c^6 - 4553660109375*b^6)*(c + (a*x + sqrt(a^2*x^2 - b))^(1/3))^(1/4)) - 21945*a*b*
c^7*((295147905179352825856*b^4*c^24 - 149294327647331942400*b^5*c^18 + 28319017190031360000*b^6*c^12 - 238742
9351424000000*b^7*c^6 + 75476916312890625*b^8)/(a^4*c^29))^(1/4)*log(-27*a^3*c^22*((295147905179352825856*b^4*
c^24 - 149294327647331942400*b^5*c^18 + 28319017190031360000*b^6*c^12 - 2387429351424000000*b^7*c^6 + 75476916
312890625*b^8)/(a^4*c^29))^(3/4) + 27*(2251799813685248*b^3*c^18 - 854268995174400*b^4*c^12 + 108028477440000*
b^5*c^6 - 4553660109375*b^6)*(c + (a*x + sqrt(a^2*x^2 - b))^(1/3))^(1/4)) - 4*(536870912*b*c^11 + 428032000*a^
2*b*c^5*x^2 - 214016000*b^2*c^5 - 2464*(131072*a*b*c^8 + 104975*a*b^2*c^2)*x - 3*(273940480*a^3*c^6*x^3 - 1174
40512*b*c^9 - 159174400*a^2*b*c^3*x^2 + 79587200*b^2*c^3 - 17765*(65536*a*b*c^6 - 6825*a*b^2)*x - 1045*(262144
*a^2*c^6*x^2 - 983040*b*c^6 - 152320*a*b*c^3*x + 116025*b^2)*sqrt(a^2*x^2 - b))*(a*x + sqrt(a^2*x^2 - b))^(2/3
) - 352*(917504*b*c^8 + 1216000*a*b*c^5*x - 734825*b^2*c^2)*sqrt(a^2*x^2 - b) - 12*(33554432*b*c^10 + 37452800
*a^2*b*c^4*x^2 - 18726400*b^2*c^4 - 385*(65536*a*b*c^7 + 62985*a*b^2*c)*x - 385*(65536*b*c^7 + 97280*a*b*c^4*x
- 62985*b^2*c)*sqrt(a^2*x^2 - b))*(a*x + sqrt(a^2*x^2 - b))^(1/3))*(c + (a*x + sqrt(a^2*x^2 - b))^(1/3))^(3/4
))/(a*b*c^7)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*x^2-b)^(1/2)/(a*x+(a^2*x^2-b)^(1/2))^(1/3)/(c+(a*x+(a^2*x^2-b)^(1/2))^(1/3))^(1/4),x, algorithm
="giac")

[Out]

Timed out

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maple [F]  time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {a^{2} x^{2}-b}}{\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{3}} \left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{3}}\right )^{\frac {1}{4}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2*x^2-b)^(1/2)/(a*x+(a^2*x^2-b)^(1/2))^(1/3)/(c+(a*x+(a^2*x^2-b)^(1/2))^(1/3))^(1/4),x)

[Out]

int((a^2*x^2-b)^(1/2)/(a*x+(a^2*x^2-b)^(1/2))^(1/3)/(c+(a*x+(a^2*x^2-b)^(1/2))^(1/3))^(1/4),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a^{2} x^{2} - b}}{{\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*x^2-b)^(1/2)/(a*x+(a^2*x^2-b)^(1/2))^(1/3)/(c+(a*x+(a^2*x^2-b)^(1/2))^(1/3))^(1/4),x, algorithm
="maxima")

[Out]

integrate(sqrt(a^2*x^2 - b)/((a*x + sqrt(a^2*x^2 - b))^(1/3)*(c + (a*x + sqrt(a^2*x^2 - b))^(1/3))^(1/4)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {a^2\,x^2-b}}{{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/3}\,{\left (c+{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/3}\right )}^{1/4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2*x^2 - b)^(1/2)/((a*x + (a^2*x^2 - b)^(1/2))^(1/3)*(c + (a*x + (a^2*x^2 - b)^(1/2))^(1/3))^(1/4)),x)

[Out]

int((a^2*x^2 - b)^(1/2)/((a*x + (a^2*x^2 - b)^(1/2))^(1/3)*(c + (a*x + (a^2*x^2 - b)^(1/2))^(1/3))^(1/4)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a^{2} x^{2} - b}}{\sqrt [4]{c + \sqrt [3]{a x + \sqrt {a^{2} x^{2} - b}}} \sqrt [3]{a x + \sqrt {a^{2} x^{2} - b}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a**2*x**2-b)**(1/2)/(a*x+(a**2*x**2-b)**(1/2))**(1/3)/(c+(a*x+(a**2*x**2-b)**(1/2))**(1/3))**(1/4),
x)

[Out]

Integral(sqrt(a**2*x**2 - b)/((c + (a*x + sqrt(a**2*x**2 - b))**(1/3))**(1/4)*(a*x + sqrt(a**2*x**2 - b))**(1/
3)), x)

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